On R¨odl’s Theorem for Cographs

A theorem of R¨odl states that for every ﬁxed F and ε > 0 there is δ = δ F ( ε ) so that every induced F -free graph contains a vertex set of size δn whose edge density is either at most ε or at least 1 − ε . R¨odl’s proof relied on the regularity lemma, hence it supplied only a tower-type bound for δ . Fox and Sudakov conjectured that δ can be made polynomial in ε , and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when F = P 4 . In fact, they show that the same conclusion holds even if G contains few copies of P 4 . In this note we give a short proof of a more general statement.


Introduction
Our investigation here is related to two of the most well studied problems in extremal graph theory.A graph-family F has the Erdős-Hajnal property if there is c = c(F) > 0 such that every n-vertex induced F-free graph has a clique of independent set of size at least cn c .The famous Erdős-Hajnal conjecture [3] states that every non-empty family of graphs F has the Erdős-Hajnal property.A variant of the Erdős-Hajnal conjecture was obtained by Rödl [9], who proved that if G is induced F -free then for every ε > 0, G contains a set of vertices of size δ F (ε) • n whose edge density is either at most ε or at least 1 − ε (we will henceforth call such sets ε-homogenous).Rödl's proof relied on Szemerédi's regularity lemma, and thus supplied very weak tower-type bounds for δ F (ε). Fox and Sudakov [5] obtained a quantitative improvement over Rödl's proof by showing that one can take δ F (ε) = ε O F (log 1/ε) .They further conjectured that the statement holds already when δ = ε O F (1) , noting that a solution of their conjecture would also resolve the Erdős-Hajnal conjecture.
Motivated by Nikiforov's [8] strengthening of Rödl's theorem, Fox, Nguyen, Scott and Seymour [4] introduced the following variant of the conjecture raised in [5].Let us say that a graph F on f vertices is viral if for every ε > 0, there is δ = ε O F (1) so that every graph G that contains at most δn f induced copies of F must contain an ε-homogenous set of size δn.The main result of [4] was that P 4 , the path on 4 vertices, is viral.Our aim in this paper is to give a very short proof of this result.In fact, we prove the following much more general statement.
Theorem 1.Let F be a finite graph family which contains a bipartite, a co-bipartite and a split graph.Suppose that F satisfies the Erdős-Hajnal property.Then for every ε > 0 there is δ = poly(ε) > 0, where the polynomial depends on F, such that the following holds.For every n-vertex graph G, n ≥ 1/δ, if G contains less than δn |V (F )| induced copies of F for every F ∈ F, then G contains an ε-homogenous set X of size |X| ≥ δn.
Since P 4 is bipartite, co-bipartite and split, and since it is well known that the Erdős-Hajnal conjecture holds for P 4 , the fact that P 4 is viral follows immediately from Theorem 1.As another example, it is well known that G is a line graphs if and only if it is induced F-free, where F is a family of 9 graphs.Since F satisfies the assertion of Theorem 1, we deduce that if G contains only poly(ε)n |V (F )| induced copies of each of the 9 graphs F ∈ F, then G contains an ε-homogenous set of size poly(ε) • n.We finally mention that we can prove a version of Theorem 1 which applies to some infinite families F is analogy with Theorem 5 in [6]; the details are omitted.
One of the tools used in [4] is the polynomial removal lemma for P 4 of [1].While our proof of Theorem 1 is inspired by an alternative proof of this result in [6], our proof here is much simpler.It is in fact very similar to the Regularity+Turán+Ramsey proof technique that was first introduced in [9] and later used in numerous works applying the regularity method.The key differences which give the improved polynomial bound are that we replace the application of Szemerédi's regularity lemma, with an application of the Alon-Fischer-Newman regularity lemma [2], and that we replace the application of Ramsey's theorem, with an application of the (assumed) Erdős-Hajnal property.

Proof of Theorem 1
We will need the following lemma.
Lemma 2.1.Let F be a family of graphs which contains a bipartite, a co-bipartite and a split graph.Then for every γ > 0 there is β = poly(γ) > 0 (where the polynomial depends on F) such that the following holds.For every k 0 ≥ 1 and every n-vertex graph G, n ≥ k 0 /β, if G contains less than βn |V (F )| induced copies of F for every F ∈ F, then G has an equipartition into k parts V 1 , . . ., V k , where k 0 ≤ k ≤ k 0 /β, such that for all but γ k 2 of the pairs Lemma 2.1 follows immediately from the conditional regularity lemma of Alon, Fischer and Newman [2] for graphs with bounded VC-dimension (see also [7]), and the fact that if F contains a bipartite graph, a co-bipartite graph and a split graph, then the family of induced F-free graphs has bounded VC-dimension (see Lemma 2.2 in [6]).
Let G be an n-vertex graph.We may assume that G contains less than βn |V (F )| induced copies of F for every F ∈ F. Let then V 1 , . . ., V k be a partition as guaranteed by Lemma 2.1.Note that k = poly(1/ε) because β = poly(ε) and k 0 = poly(1/ε).Define an auxiliary graph Case 1: R contains an induced copy of some F ∈ F. Let i 1 , . . ., i m ∈ A be the vertices of such a copy; so m = |V (F )| ≤ f .Sample v i j ∈ V i j uniformly at random and independently, j = 1, . . ., m.By the definition of R, for each i j i ℓ ∈ E(F ) we have d(V i j , V i ℓ ) ≥ 1 − γ, and for each i j i ℓ / ∈ E(F ) we have d(V i j , V i ℓ ) ≤ γ.By the union bound, the probability that v i 1 , . . ., v im do not span an induced copy of F is at most m Case 2: R is induced F-free.By the choice of c = c(F), the graph R contains a clique or independent set B ⊆ A of size |B| ≥ c|A| c ≥ c( 1 2γ ) c ≥ 4/ε, using our choice of γ.Suppose without loss of generality that B is an independent set, and write B = {i 1 , . . ., i t }.For every 1 ≤ j < ℓ ≤ t, we have d(V i j , V i ℓ ) ≤ γ ≤ ε 4 .Also, the number of edges which are contained in one of the sets V i 1 , . . ., V it is at most t • n/k 2 ≤ tn 2 2k 2 ≤ εt 2 n 2 8k 2 , using that t ≥ 4/ε.Hence, setting X = V i 1 ∪ • • • ∪ V it , we have |X| = t n k = poly(ε)n and